The above results are obtained from the Bayesian parameter estimation in [1]} by comparing the modified waveform with the GW strain data of 12 GW events in LIGO-Virgo O1/O2 catalog. Here we would like to mention here that the amplitude birefringence also affects the statistical distributions of \(\cos {\iota }\)  over the population of binary black hole merges [2]} with \(\iota \)  being the inclination angle of the binary black hole system. One can consider the universe is homogeneous and isotropic at cosmological scales, and that gravitational physics does not have any preferred direction. This implies that the underlying distribution for \(\cos {\iota }\)  is flat, meaning its distribution is symmetric about zero when the amplitude birefringence is absent. When the amplitude birefringence induced by \({\cal G}_1\)  is included, the distribution of \(\cos \iota \)  will preferentially have \(\cos \iota >0 (<0)\)  if \({\cal G}_1 <0 (>0)\)  [2]}. By checking the statistic distribution of \(\cos \iota \)  for GW events in GWTC-2 catalog, it is able to impose a constraint on \({\cal G}_1\)  [2]}, i..e,
\(|{\cal G}_1| \lesssim 1000\; {\rm km}.\) 
